Experimental probability

Activities to try

1. 1

   Drop a drawing pin and watch the estimate settle

   Why it works: When a device is biased or its structure is unknown there is no formula to reach for — you must run trials. The relative frequency you read off IS your best estimate of a probability you could not calculate from symmetry, and that estimate earns its reliability from the number of trials: a 50-drop figure is rough and jumpy, while a settled line after many hundreds or thousands of trials is much more reliable, though still an estimate.

   1. Take a drawing pin (or a visibly bent coin) whose true point-up probability cannot be worked out from symmetry, and a frequency table with a relative-frequency (f ÷ n) column.
   2. Drop the pin 50 times, recording point-up/point-down tallies, then compute relative frequency = point-up frequency ÷ 50.
   3. State this relative frequency as your best estimate of the probability, and note explicitly that there is no theoretical value to calculate from symmetry because the pin is not symmetric.
   4. Reach a much larger number of trials and recompute the relative frequency: pool the whole class's physical drops (same type of pin, dropped the same way onto the same kind of surface) to get into the hundreds, and/or use a hidden-true-probability pin/coin simulator (with the same frequency table and relative-frequency line) to reach 500 then 5000 trials quickly. Note that the simulator's settled value estimates the simulator's hidden probability, not your own pin's. Watch how the value jumps around early and then tends to settle near a steady figure.
   5. Decide how much you would trust the 50-drop figure versus the large-trials figure, and say why.
2. 2

   Fair-device check: experimental versus theoretical on one scale

   Why it works: Where a theoretical value does exist, relative frequency can be compared with it: discrepancies in a short run are normal variation, and the experimental value tends toward the theoretical as trials increase. This frames why, for a biased device with no symmetry-derived value, the relative frequency is all you have.

   1. Toss an ordinary coin (or roll a fair die), record outcomes in a frequency table, and compute the relative frequency for each outcome.
   2. Mark both your experimental relative frequency and the theoretical probability on the same 0-to-1 scale.
   3. Explain a small gap between them as expected short-run variation rather than a mistake or evidence that the coin is biased.
   4. Repeat with more tosses (or pool data) and see whether the experimental value tends to settle closer to the theoretical one.
3. 3

   Compute and judge reliability from given tables

   Why it works: Computing relative frequency from a table and judging sample-size reliability are skills that can be practised and checked — but they sit on top of the run-and-stabilise experience, which is where the meaning of 'more trials → better estimate' actually comes from.

   1. Given a completed frequency table, fill in the relative-frequency column (each outcome's frequency ÷ total trials).
   2. Decide, for a bent coin, which of two samples (e.g. 30 trials versus 800 trials) gives the more reliable estimate of its bias, and justify your choice.
   3. Explain in a sentence why you cannot calculate the bent coin's probability from symmetry.